BDH-CQ: In-Context Learning with Recurrent Latent Reasoning

Björn Engdahl; Adrian Kosowski; Jan Chorowski; Zuzanna Stamirowska; Przemysław Uznański; Junlin Jiang; Rohan Phadke; Remigiusz Kinas; Richard Zhong

Source-linked AI summary

BDH-CQ: In-Context Learning with Recurrent Latent Reasoning

Björn Engdahl, Adrian Kosowski, Jan Chorowski, Zuzanna Stamirowska, Przemysław Uznański, Junlin Jiang, Rohan Phadke, Remigiusz Kinas, Richard Zhong

arXiv:2608.09888v1Aug 10, 2026cs.NEcs.AIcs.LGstat.ML

TL;DR

Existing approaches have largely separated in-context learning from latent iterative reasoning, leaving limited evidence that an unseen transformation can be learned from demonstrations without verbalized intermediate steps. BDH-CQ combines recurrent-memory updates with continuous latent computation and reaches 29.5% pass@2 at $0.00070 per ARC-AGI-1 task, beyond the reported cost–accuracy frontier.

Problem
Latent reasoning and in-context learning have largely developed separately, with limited evidence that unseen transformations can be acquired solely from inference-time demonstrations.
Method
BDH-CQ combines evolving recurrent memory for in-context learning with iterative reasoning in a structured continuous latent space, evaluated through ARC and controlled interventions.
Results
29.5% pass@2 at $0.00070 per task on ARC-AGI-1 breaks through the previously reported cost–accuracy Pareto frontier.
Takeaways & Limitations
Controlled experiments show reliable binding of new mappings, extrapolation of several operators, benefits from demonstration coverage, and structured limits under composition.
Takeaways & Limitations
Dimensions, exact update rules, and implementation details remain proprietary, limiting reproducibility of the system’s internal mechanisms.

Abstract

from arXiv · show

We introduce BDH-CQ, a reasoning model that combines in-context learning with recurrent latent reasoning. Inputs presented at inference time continuously update the model's recurrent memory; the model then solves a query through iterative computation in a high-dimensional latent space, without verbalizing its intermediate reasoning. We evaluate the model on the public ARC-AGI-1 evaluation set and use controlled ARC-like interventions to study what it learns from demonstrations, how consistently it applies an inferred transformation, and which concepts remain difficult. A 150M-parameter configuration reaches 29.5% pass@2 at a computed inference cost of \$0.0007 per task. This operating point breaks through the previously reported ARC-AGI-1 cost-accuracy Pareto frontier, establishing a new state of the art in benchmark cost efficiency.

1 Introduction

BDH-CQ combines demonstration-driven recurrent memory with iterative latent reasoning, solving unseen ARC tasks without decoding intermediate reasoning into language. On ARC-AGI-1, its 150M-parameter configuration reaches 29.5% pass@2 at a computed $0.00070 per task, breaking through the previously reported cost–accuracy Pareto frontier.

System: Latent reasoning repeatedly transforms continuous hidden states and decodes only the answer, avoiding natural-language tokens as the mandatory representation of internal computation.This regime can preserve partial hypotheses or candidate transformations without serializing each one.
Motivation: BDH-CQ addresses the separation between flexible in-context learning and compact latent-space recursive solvers that rely on optimization and task-specific identities.The introduction positions the system as combining capabilities that had largely developed separately.
System: BDH-CQ combines in-context learning through evolving recurrent memory with iterative reasoning in a structured continuous latent space.Demonstrations update recurrent memory, after which the query is solved through iterative computation in a high-dimensional latent workspace; intermediate reasoning states are not decoded into language.
Results: 29.5% pass@2 at a computed $0.00070 per task establishes a new state-of-the-art point on the public ARC-AGI-1 cost–accuracy frontier.This result comes from a 150M-parameter configuration; neither task identifiers nor evaluation-task demonstration pairs participate in training, and no parameters are updated at inference time.
Evaluation: ARC serves as both evaluation and experimental substrate because sparse demonstrations specify new visual transformations and exact outputs make learning and rule consistency directly inspectable.Controlled experiments study what demonstration-conditioned operator schemas can bind and examine transformation capabilities and failure modes across concept-organized families.

2 ARC as a controlled, verifiable testbed for BDH-CQ’s in-context generalization

ARC tests in-context generalization by requiring systems to infer new transformations from a handful of demonstrations and apply them to novel inputs. Its compact colored grids support controlled study of latent computation, exact execution, consistency, and inference cost with visually inspectable outcomes.

ARC as a controlled, verifiable testbed for BDH-CQ’s in-context generalization: ARC requires inferring objects, relations, and operations from a handful of demonstrations, then applying the resulting transformation rule to a new input.The benchmark was designed to study how much experience and prior structure a system needs to acquire a skill.
ARC as a controlled, verifiable testbed for BDH-CQ’s in-context generalization: ARC tasks require little factual knowledge or linguistic fluency while expressing object relations, counting, symmetry, topology, spatial transformations, and compositions.Each answer is exactly verifiable, failures are visually inspectable, and multiple test inputs expose whether a rule is applied consistently.
ARC as a controlled, verifiable testbed for BDH-CQ’s in-context generalization: In the illustrated task, the inferred rule marks cells occupied in both binary panels and transfers that intersection relation to the query.The input does not name intersection, alignment, or the meaning of the output color, so the transformation must be inferred from demonstrations.
ARC as a controlled, verifiable testbed for BDH-CQ’s in-context generalization: ARC provides a controlled visual language for studying learning from context, latent computation, exact execution, and inference cost together.Its tasks are compact, exact, and easy to inspect while requiring transformations to be inferred from examples.

3 Introducing BDH-CQ: In-context learning through recurrent memory and latent reasoning

BDH-CQ extends the BDH architecture into a reasoning system that learns visual transformations from demonstrations through recurrent memory, then solves queries via iterative computation in a structured latent workspace. Its contextual memory supports in-context learning, while the reasoning workspace carries computation for the current query.

System design: BDH-CQ combines a structured latent workspace and recurrent computation over model depth with an interface for learning visual transformations from demonstrations.It extends BDH recurrent systems previously used for constraint satisfaction.
Recurrent contextual memory: The recurrent memory processes demonstration elements sequentially, updating as each demonstration’s content is incorporated while model parameters remain fixed.Its update is expressed as S_t = U_θ(S_{t−1}, D_t).
Recurrent contextual memory: Information available to later inputs depends on associations accumulated from earlier inputs and outputs, without a growing explicit key–value cache.The contextual state is analogous to context-dependent associations formed by attention, fast-weight memory, and linear-attention mechanisms.
Latent reasoning: After ingesting the demonstrations and query, BDH-CQ encodes the query, iterates computation in a structured latent workspace, and decodes the final output.The process is represented by H_0 = E_θ(x⋆, S_K), H_{r+1} = F_θ(H_r, S_K), and ŷ = G_θ(H_R).
System interface: The contextual memory S_t changes as evidence is encountered and supports in-context learning, whereas the reasoning workspace H_r carries the ongoing computation for the current query.Dimensions, exact update rules, and implementation details remain proprietary.

4 Training data and objective

Training uses ARC tasks with demonstration and test pairs, teaching the model to incorporate preceding examples into recurrent context and produce exact target grids. The 150M-parameter model is trained on curated ARC-style data combining private and public sources, with additional augmentations.

Each ARC task contains K_i demonstration pairs and Q_i test pairs, with Q_i ≥ 1.
The objective predicts outputs after preceding examples are incorporated into recurrent context, targeting exact output grids.The complete internal training recipe remains proprietary.
The 150M-parameter model trains on privately curated and publicly available ARC-style datasets, with additional augmentations increasing data variety.Public sources include ARC-AGI-1, RE-ARC, ConceptARC, ARC-Heavy, and ARC-GEN100K.

5 Public ARC-AGI-1 evaluation

On the 400-task public ARC-AGI-1 evaluation split, the default 150M-parameter system achieves 29.5% pass@2 at a computed cost of $0.00070 per task, beyond the previously reported cost–accuracy Pareto frontier. An independent black-box audit reproduced the deployed system’s 29.5% pass@2 score.

Default evaluation: 29.5% pass@2 is achieved by the default 150M-parameter system at approximately 0.85 H200 GPU-seconds per task.The default system follows the leaderboard’s two-attempt convention, producing up to two ranked candidates and scoring if either is correct.
Cost efficiency: $0.00070 per task is the computed cost at $3 per H200-hour, less than one-tenth of a cent.The dollar value is computed from measured hardware time for the leaderboard’s score–cost plane.
Cost efficiency: The 29.5% pass@2 point lies beyond the previously reported Pareto frontier, establishing a new state of the art in ARC-AGI-1 cost efficiency.No plotted system attains at least this accuracy at equal or lower reported cost.
Independent evaluation: 29.5% pass@2 is reproduced by an independent black-box audit of the deployed system on the public ARC-AGI-1 evaluation set.The auditors used a documented protocol without access to model weights.

6 Behavioral analysis on ARC-like tasks

BDH-CQ shows strong but uneven ARC-like generalization: it applies propagation and copying reliably, while ordering, nesting, and some operation compositions expose distinct failure modes. ConceptARC results also reveal a substantial gap between solving individual test inputs and consistently solving whole tasks.

ConceptARC evaluation: ConceptARC uses 16 concept families and strict task accuracy, with test-pair accuracy reported to separate consistent transformation application from individual-input success.A task is correct only when all three test inputs are solved; each family contains ten tasks and thirty test inputs.
ConceptARC evaluation: 9/10 pass@2 was achieved by ExtendToBoundary, FilledNotFilled, and TopBottom2D, versus 2/10 for Copy and Order.The paper cautions that ten tasks per family make these descriptive profiles rather than reliable rankings.
ConceptARC evaluation: 19/30 semantic test-pair pass@2 on Copy contrasted with 2/10 strict tasks, showing many correct inputs without consistent task-wide transformation application.In the opaque replication, 20/30 pairs were correct while only 2/10 tasks were solved.
Controlled ARC-like interventions: 48/48 held-out outputs remained correct for propagation across distances 2–8 and copying across one to four target sites, while ordering fell to 29/36 at length six, 8/24 at seven, and 1/24 at eight.Propagation and copying did not reach a ceiling in the tested range; ordering was nearly saturated only through five objects.
Controlled ARC-like interventions: At ordering length eight, only 3/24 outputs had correct dimensions, whereas all 36 depth-five nesting outputs had correct dimensions and mean best-candidate cell accuracy exceeded 99.9%.Ordering failures disrupted output construction, while nesting errors typically preserved structure and differed in one containment decision.
Operation composition: 72/72 outputs were solved for atomic reflection, atomic rotation, relocation, and rotation-plus-relocation, but reflection-plus-relocation reached 47/72 and color-swap-plus-relocation 0/72.Color swap was acquired atomically only in the original family, at 26/72 pooled.

7 Scaling reasoning effort increases pass@2

The section evaluates how latent reasoning effort affects pass@2 and cost, finding that increasing effort from LOW to MEDIUM to HIGH boosts pass@2.

Scaling reasoning effort increases pass@2: Increasing latent reasoning effort from LOW to MEDIUM and from MEDIUM to HIGH boosts pass@2.The model is trained across different reasoning-effort levels, and the inference setting selects the applied effort.
Scaling reasoning effort increases pass@2: The evaluation compares pass@2 and cost across LOW, MEDIUM, and HIGH reasoning efforts.Table 5 and Figure 7 report how pass@2 and cost scale with reasoning effort.

8 Related work

Prior work spans verbalized, continuous, compressed, and communicated latent reasoning, alongside recurrent-depth architectures and task-trained recursive solvers. BDH-CQ combines contextual flexibility with a continuous recurrent workspace, while using ConceptARC’s ontology to organize capability analysis.

Verbalized reasoning: BDH-CQ replaces natural-language intermediate reasoning with a continuous recurrent workspace while retaining the contextual flexibility of chain-of-thought prompting.Chain-of-thought combines in-context specification with an autoregressive natural-language scratchpad; BDH-CQ makes intermediate language generation unnecessary.
Continuous thoughts in language models: Coconut feeds a Transformer’s previous final hidden state into the next input embedding and progressively replaces verbal chain-of-thought steps with continuous thoughts.The passage identifies Coconut as a seminal demonstration of continuous hidden-state feedback and notes that one state can preserve multiple candidate continuations.
Continuous thoughts in language models: Continuous-thought theory shows that one state can carry multiple active hypotheses, while latent iteration can exploit parallel computational structure.The cited construction concerns directed-graph reachability and describes multiple search frontiers being encoded and expanded in parallel.
Compressed discrete and communicated latent states: 11.6-fold fewer reasoning tokens is reported for Abstract-CoT, which uses a short autoregressive sequence from a learned reserved vocabulary instead of verbal rationales.The passage places Abstract-CoT at the discrete end of nonlinguistic reasoning and distinguishes it from latent communication between agents.
Recurrent depth and looped models: Recurrent-depth models reuse shared computation across iterations, while HRM and TRM recursively update latent and answer states but use a transductive ARC pipeline.The ARC pipeline augments demonstration pairs during optimization, assigns learned identity embeddings, and votes over augmentations; ConceptARC instead organizes ARC-like tasks by spatial and semantic concepts.

9 Discussion

BDH-CQ achieves 29.5% pass@2 on ARC-AGI-1 at $0.00070 per task, breaking the reported cost–accuracy Pareto frontier. Controlled studies characterize its contextual capabilities and limits, while the architecture motivates scaling and broader latent-reasoning applications.

System-level result: 29.5% pass@2 at $0.00070 per task establishes BDH-CQ as a new state of the art in benchmark cost efficiency.This result comes from a 150M-parameter configuration on the public ARC-AGI-1 evaluation set.
Controlled capabilities and limits: BDH-CQ applies dense task-specific color mappings to all 96 held-out outputs, while boundary propagation and copying extrapolate throughout tested ranges.Ordering and nested containment reveal distinct limits; matched demonstrations eliminate depth-five nesting failures and substantially improve length-eight ordering.
Capability profile: The ConceptARC ontology organizes BDH-CQ’s learned capability profile, while post-freeze generators localize boundaries in consistency, composition, conditional execution, and output construction.The study establishes this account for ARC-like visual reasoning.
Future directions: BDH-CQ points toward reasoning systems that unify memory, adaptation, and inference, with future work targeting larger models, longer training runs, and ARC-AGI-2.Other proposed tests include Sudoku, language, mathematical reasoning, output construction, conditional binding, demonstration coverage, and multi-operator composition.
Scaling: BDH-CQ scales to large models through inherited tensor sharding, with early pretraining experiments confirming Transformer-like scaling laws from 1B to 600B parameters.These experiments preserve latent reasoning capabilities specific to BDH-CQ.

10 Conclusion

BDH-CQ combines in-context learning with recurrent latent reasoning in a compact system, updating recurrent memory from demonstrations and solving tasks through iterative continuous computation. On ARC-AGI-1, it reaches 29.5% pass@2 at $0.00070 per task, establishing a new state of the art in benchmark cost efficiency.

10 Conclusion: 29.5% pass@2 at $0.00070 per task establishes a new state of the art in benchmark cost efficiency.The result is reported on ARC-AGI-1.
10 Conclusion: BDH-CQ combines in-context learning and recurrent latent reasoning in a compact, practical system.The model’s design integrates both capabilities rather than treating them as separate mechanisms.
10 Conclusion: Demonstrations modify recurrent memory at inference time, after which the task is solved through iterative continuous computation without verbalized chain-of-thought reasoning.The reasoning occurs in the model’s latent computation rather than through an explicit textual rationale.

A Evaluation set analysis … A.4 Failure structure and scope

The evaluation analyses find that surface descriptors weakly predict success, while mechanic groups and controlled ladders expose substantial, task-construction-sensitive limitations in transformation selection and parameterization. Failures often preserve dimensions and localize cell differences, but output-only analysis cannot reveal the latent rule, and generated-set conclusions have known scope limitations.

A.1 Evaluation sets and surface descriptors: 39 public-set bucketings and 38 calibrated-generated-set bucketings show weak surface explanations; grid size is strongest but has pseudo-R2 = 0.072 on public tasks and 0.010 on generated tasks.Nine public-set bucketings exceed the permutation-search null, while none does so on the calibrated generated set.
A.2 Transformation mechanics: 1,131 approximately balanced tasks across 16 mechanics reveal solve rates spanning 65.7 percentage points, from 68.6% for flood fill to 2.9% for gravity and stacking.The observed spread exceeds the 95th-percentile permutation spread of 27.0 points, with no matching permutation (p < 0.00005).
A.2 Transformation mechanics: ρ = 0.300 across 16 mechanics shows no reliable public/generated rank correspondence, and hand-written gravity tasks score 80–100% versus 2.9% for generated gravity tasks.Generated mechanics were authored by GPT-5.6 from prompts naming the target mechanic, so rates also reflect task instantiation.
A.3 Controlled structural manipulations: 68/120 genuine conditional-selection tasks are solved, or 56.7% (95% CI [48, 65]), a decrease of 43.3 points from the 40/40 control.The control indicates that the cost attaches to using a cue to select between rules rather than merely processing the cue.
A.3 Controlled structural manipulations: 12/40 tasks are solved when the test value is demonstrated, versus 0/120 when it is absent, including 0/40 interpolation and 0/80 extrapolation conditions.The result identifies demonstration presence, rather than interpolation versus extrapolation, as the relevant boundary within this family.
A.3 Controlled structural manipulations: 26/40 opposite-corner panel unions are solved versus 1/40 with three panels, while touching two-panel unions score 3/40.Answer color and expected union density are held fixed; the pattern is consistent with, but does not directly test, a segmentation limitation.
A.4 Failure structure and scope: 400-task failure comparisons find 251 calibrated-generated failures and 282 public failures; most preserve output dimensions, and many shape-correct failures differ in few cells.These are output-level signatures and do not identify the latent rule, because incomplete application and narrower-rule application can be observationally indistinguishable.
A.4 Failure structure and scope: 82.9% agreement with requested mechanic labels on a labeled 659-task subset, unknown prevalence of contradictory test outputs, and 24% deduplication constrain generated-set interpretation.Family differences for conditional selection, unseen values, panel number/separation, and support-chain length have not yet been shown to generalize across visual operations.